Editing Work (physics) (section)

===Torque and rotation===
A [[Couple (mechanics)|force couple]] results from equal and opposite forces, acting on two different points of a rigid body.  The sum (resultant) of these forces may cancel, but their effect on the body is the couple or torque '''T'''.  The work of the torque is calculated as
<math display="block"> \delta W = \mathbf{T} \cdot \boldsymbol{\omega} \, dt,</math>
where the {{math|'''T''' ⋅ '''''ω'''''}} is the power over the instant {{math|''dt''}}.  The sum of these small amounts of work over the trajectory of the rigid body yields the work,
<math display="block"> W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt.</math>
This integral is computed along the trajectory of the rigid body with an angular velocity {{math|'''''ω'''''}} that varies with time, and is therefore said to be ''path dependent''.

If the angular velocity vector maintains a constant direction, then it takes the form,
<math display="block"> \boldsymbol{\omega} = \dot{\phi}\mathbf{S},</math>
where <math>\phi</math> is the angle of rotation about the constant unit vector {{math|'''S'''}}.  In this case, the work of the torque becomes,
<math display="block">W = \int_{t_1}^{t_2} \mathbf{T} \cdot \boldsymbol{\omega} \, dt = \int_{t_1}^{t_2} \mathbf{T} \cdot \mathbf{S} \frac{d\phi}{dt} dt = \int_C\mathbf{T}\cdot \mathbf{S} \, d\phi,</math>
where {{math|''C''}} is the trajectory from <math>\phi (t_{1})</math> to <math>\phi (t_{2})</math>.  This integral depends on the rotational trajectory <math>\phi (t)</math>, and is therefore path-dependent.

If the torque <math>\tau</math> is aligned with the angular velocity vector so that,
<math display="block" qid=Q48103> \mathbf{T} = \tau \mathbf{S},</math>
and both the torque and angular velocity are constant, then the work takes the form,<ref name="Young">{{cite book | author1 = Hugh D. Young |author2 = Roger A. Freedman | name-list-style=amp | title = University Physics | edition = 12th | publisher = Addison-Wesley | year = 2008| isbn = 978-0-321-50130-1| page = 329}}</ref>
<math display="block">W = \int_{t_1}^{t_2} \tau \dot{\phi} \, dt = \tau(\phi_2 - \phi_1).</math>
[[File:Work on lever arm.png|250px|thumb|right|alt=Work on lever arm|A force of constant magnitude and perpendicular to the lever arm]]

This result can be understood more simply by considering the torque as arising from a force of constant magnitude {{math|''F''}}, being applied perpendicularly to a lever arm at a distance <math>r</math>, as shown in the figure.  This force will act through the distance along the circular arc <math>l=s=r\phi</math>, so the work done is
<math display="block"> W = F s = F r \phi .</math>
Introduce the torque {{math|1=''τ'' = ''Fr''}}, to obtain
<math display="block"> W = F r \phi = \tau \phi ,</math>
as presented above.

Notice that only the component of torque in the direction of the angular velocity vector contributes to the work.